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# SQA Higher Maths Integration of exponentials and logarithms

To find the integral of equations or variables in the form eˣ, use the following rule.

Integrate variables in the form aˣ using the exponentials rule. Integrate 2 functions multiplied together by parts; for u, choose the function that is simpler when differentiated.

Integrate variables in the form lnx using integration by parts: ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. Let dv/dx = 1 and u = the variable in the form lnx.

Integrate logarithmic functions using integration by parts: ∫ u (dv/dx) dx = uv − ∫ v (du/dx) dx. Let dv/dx = 1 and u = the logarithmic function.

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Find ∫ e³ˣ ⁻ ¹ dx.

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Integrate (eˣ − 2)²/e²ˣ dx.

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Integrate 2ˣ dx.

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# ✅

Find ∫ xe²ˣ dx.

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The diagram shows part of the curve y = −4²ˣ ⁻ ⁸ + 5. Find the exact value of the volume of the solid formed when the shaded region is rotated completely about the x−axis.

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# ✅

Find the integral ∫ 3lnx dx.

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The diagram shows part of the curve y = −ln(x + 1) − 2x² + 8. Find the exact value of the volume of the solid formed when the shaded region is rotated completely about the x−axis.

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# ✅

Integrate ∫ x(lnx)² dx.

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# ✅

Integrate xlog₅(4x) dx.

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# ✅

What is ∫ log₃(4x + 8) dx?

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